If z1 = z1 = 1 + i and z2 = 1 - i are two complex
numbers. Show that their quotient is also a complex number.

Solution:

The quotient
which is also a complex number. (Proved)

Any integral power of a complex number is a complex number i.e.,
any integral power of a complex number can be expressed in the form A + iB where A
and B are real.Let z = x + iy be a complex number where x, y are real.

If n is a positive integer then, zn = z.z.z.....to n factors

= (x + iy)(x + iy).....to n factors

= A + iB

(Since the product of more than two complex numbers is also a complex number),
where A and B are real.

Now if n is a negative integer, let n = -m (where m is a positive integer), then

Hence any integral power of a complex number is a complex number.

Example:

If z = 1 + i and n = 2 then show that zn is also a complex number.

Solution:

zn = z2 = (1 + i)(1 + i) = (1 + i)2 = 1 + 2i +
i2

= 1 + 2i - 1 = 2i which is also a complex number. (Proved)

Any root of a complex numbers is a complex number i.e., any root of a
complex number can be expressed in the form A + iB where A and B are real.

Hence it is clear that mn is real when m is purely real quantity and mn is either purely
real or purely imaginary quantity when m is purely imaginary quantity.

Again x ≠ 0, y ≠ 0, hence equation (1) is satisfied if and only if m is an
imaginary number of the form A + iB where A ≠ 0 and B ≠ 0 are real.

Hence any root of a complex numbers is a complex number.

Example:

If z = 2 + i and n = 3 then show that n√z is also a complex number.

Solution:

Let n√z = m ⇒ 3√z = m

⇒ 3√2+ i = m

⇒ m = 3√2+ i

Hence m is also an imaginary number of the form A + iB.

Hence any root of z is also a complex number. (Proved)

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